Does $\tan x\cdot \cos x$ equal $\sin x$? Is it true that $\tan x\cdot \cos x = \sin x$? If I put $x=30$ in my calculator then I don't get the same answer as $\sin 30$, why is this? Don't the two cosines cancel out? I'm probably missing something really stupid here.
 A: By definition,
$$
\tan x=\frac{\sin x}{\cos x}
$$
but in order to give sense to this definition, the $\cos$ must be nonzero, i.e. $x\neq\frac{\pi}2+k\pi$. Hence $\tan$ is defined only for $x\in\Bbb R\setminus \{\frac{\pi}{2}+k\pi\}$. For these values you can multiply both sides for $\cos$, obtaining $\tan x\cos x=\sin x$.
But pay attention: this holds only for the values I wrote! Otherwise $\tan$ is not even defined, so it wouldn't make sense to consider the relation.
Now in your case, $x=30°$ is the same as $x=\pi/6$ in radiant and in this value the $\tan$ is defined. So the relation holds.
You must had a bad work with your calculator, that's all.
A: As tan(x)≡  Sin(x)/Cos(x), you are right in that Tan(x) * cos(x) ≡ Sin(x). This is by definition of the Tan function, which is defined as Sin(x) / Cos(x). If it helps consider the right angle triangle from the unit circle, where cos(x) = Hypotenuse / adjacent and Sin(x) = opposite / hypotenuse, so Tan(x) as equalling opposite / adjacent is easy to see by dividing one by the other since the hypotenuse will cancel in both.  However, it is worth remembering that for certain values, where x is either an odd multiple of 90 degrees or odd multiples of Pi/2 if dealing in radians, this simply does not work as Cos(x) = 0 ( it is easy enough to see that at 90 degrees, this leads to 1 = 0 which is obviously nonsensical).
However, it is easy enough to prove that since Sin(30) = 0.5 and Cos(30) = root(3) /2, then tan(30) = 1 / root(3). Multiplying this by cos(30) does in fact lead us to 0.5 again, so this is right here (luckily the normal rules of communativity apply here.) However, I can see how with approximation in the calculator it would not work. If you try using the exact answers this should be proven to be true.
A: Without doubt,
$$\tan30°\cdot\cos30°=\frac1{\sqrt3}\cdot\frac{\sqrt3}2=\frac12=\sin30°$$
$$0.57735026919\cdots\times0.86602540378\cdots=0.5$$
